Noob question-Site moving.

hello,
i realy hope someone can help - im in despair,.. i am a Cpanel user(not host) im just like a cpanel admin on my site,.
Anyhow ive spent months building my site only for my current host to become unnaceptably slow - so im in the process of moving my site - its a total so far of only about 400mb-
Ive manualy downloaded the files for my site via FTP,.and uploaded them via FTP to my new server but i honestly cant work out how to move my MYsql databases ( 2 of them) to my new completely different host.
Ive built up quite a lovely member base and i cant re-create everything expecting them all to re-register etc.
Ive looked everywhere on knowledge on this and no-one seems to have the time to advise
i am not very familiar with "phpmyadmin" altho i do know how to create databases thats about it.
Could anyone possibly spare 5 mins to explain or point me in the right direction please ?
Both my old site and new site are both CPANEL :D